The coefficient `x^5` in the expansion of `(2 - x + 3x^2)^6` is

Options

A. -4692
B. 4692
C. 2346
D. -5052

Answer

To find the coefficient of \( x^5 \) in the expansion of \( (2 - x + 3x^2)^6 \), we can use the multinomial expansion. Here’s a step-by-step solution:

### Step 1: Identify the terms in the multinomial expansion
The expression \( (2 - x + 3x^2)^6 \) can be expanded using the multinomial theorem, which states that:
\[ (a_1 + a_2 + a_3)^n = \sum \frac{n!}{k_1! k_2! k_3!} a_1^{k_1} a_2^{k_2} a_3^{k_3} \]
where \( k_1 + k_2 + k_3 = n \).

In our case, \( a_1 = 2 \), \( a_2 = -x \), and \( a_3 = 3x^2 \), and \( n = 6 \).

### Step 2: Set up the equation for \( k_1, k_2, k_3 \)
We need to find combinations of \( k_1, k_2, k_3 \) such that:
\[ k_1 + k_2 + k_3 = 6 \]
and the total power of \( x \) is 5:
\[ k_2 + 2k_3 = 5 \]

### Step 3: Solve the equations
From the equations:
1. \( k_1 + k_2 + k_3 = 6 \)
2. \( k_2 + 2k_3 = 5 \)

We can express \( k_1 \) in terms of \( k_2 \) and \( k_3 \):
\[ k_1 = 6 - k_2 - k_3 \]

Substituting \( k_1 \) into the second equation:
\[ k_2 + 2k_3 = 5 \]

Now we can try different values for \( k_3 \) and find corresponding \( k_2 \) and \( k_1 \).

### Step 4: Find valid combinations
1. **If \( k_3 = 0 \)**:
- \( k_2 + 2(0) = 5 \) → \( k_2 = 5 \)
- \( k_1 = 6 - 5 - 0 = 1 \) → Combination: \( (1, 5, 0) \)

2. **If \( k_3 = 1 \)**:
- \( k_2 + 2(1) = 5 \) → \( k_2 = 3 \)
- \( k_1 = 6 - 3 - 1 = 2 \) → Combination: \( (2, 3, 1) \)

3. **If \( k_3 = 2 \)**:
- \( k_2 + 2(2) = 5 \) → \( k_2 = 1 \)
- \( k_1 = 6 - 1 - 2 = 3 \) → Combination: \( (3, 1, 2) \)

4. **If \( k_3 = 3 \)**:
- \( k_2 + 2(3) = 5 \) → \( k_2 = -1 \) (not valid)

Thus, the valid combinations are \( (1, 5, 0) \), \( (2, 3, 1) \), and \( (3, 1, 2) \).

### Step 5: Calculate the coefficients for each valid combination
1. **For \( (1, 5, 0) \)**:
\[ \text{Coefficient} = \frac{6!}{1!5!0!} (2^1)(-1)^5(3^0) = 6 \cdot 2 \cdot (-1) = -12 \]

2. **For \( (2, 3, 1) \)**:
\[ \text{Coefficient} = \frac{6!}{2!3!1!} (2^2)(-1)^3(3^1) = 20 \cdot 4 \cdot (-1) \cdot 3 = -240 \]

3. **For \( (3, 1, 2) \)**:
\[ \text{Coefficient} = \frac{6!}{3!1!2!} (2^3)(-1)^1(3^2) = 60 \cdot 8 \cdot (-1) \cdot 9 = -4320 \]

### Step 6: Sum the coefficients
Now, we sum the coefficients from all valid combinations:
\[ -12 - 240 - 4320 = -4572 \]

### Final Answer
The coefficient of \( x^5 \) in the expansion of \( (2 - x + 3x^2)^6 \) is \( -4572 \).

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